By way of contradiction, suppose that \(\Sigma \)  does not have a non-displaceable subsurface, \(\operatorname{FMap} (\Sigma )\) 
does not contain an essential shift, and that \(\operatorname{FMap} (\Sigma )\)  is not CB.
Then by assumption, \(\Sigma \)  is an avenue surface.
By Lemma REF ,
\(\mathcal {M}(E)=\lbrace x_1, x_2\rbrace \)  and for any other \(x \in E - \lbrace x_1, x_2\rbrace \) , \(E(z)\)  accumulates to
both \(x_1\)  and \(x_2\) . Consider an end \(z \in \mathcal {M}(E-\lbrace x_1, x_2\rbrace )\) , that is, an end that is
maximal in \(E-\lbrace x_1, x_2\rbrace \) . If \(E(z)\)  is countable, then \(E\)  is two-sided with
\(X=\lbrace x_1\rbrace \)  and \(Y=\lbrace x_2\rbrace \) . This cannot happen since we are
assuming there are no essential shifts (see Theorem REF ).
Therefore, \(E(z)\)  is a Cantor set.
Furthermore, either \(\Sigma \)  has genus zero or it is infinite genus
and \(\lbrace x_1, x_2 \rbrace  \subset E^G\)  (otherwise, \(\operatorname{Map} (\Sigma )\)  is not even locally CB, see
[1]}). We now show that these assumptions imply
that \(\Sigma \)  is telescoping and, by Theorem REF , \(\operatorname{Map} (\Sigma )\)  is CB, which will prove our claim.
We check the definition of telescoping. For \(i=1,2\) , let \(V_i\)  be disjoint stable neighborhoods
of \(x_i\)  such that \(E-(V_1 \cup V_2)\)  contains every maximal type in \(E-\lbrace x_1, x_2\rbrace \) .
Let \(W_i\)  be the given smaller neighborhoods of \(x_i\) . Since the \(V_i\)  are stable neighborhoods,
we have that \(V_i\)  is homeomorphic to \(W_i\) . We claim that \(E-(V_1 \cup V_2)\)  is homeomorphic to
\(E-(V_1 \cup W_2)\) . Since all maximal types in \(E-\lbrace x_1, x_2\rbrace \)  are present in \(E-(V_1 \cup V_2)\) ,
the sets \(E-(V_1 \cup V_2)\)  and \(E-(V_1 \cup W_2)\)  have the same maximal types. But all
these types are Cantor sets. Therefore, for every \(y \in V_2-W_2\)  there is a \(z \in E-(V_1 \cup V_2)\) 
such that \(E(y)\)  accumulates to \(z\) . By [1]}, there are small neighborhoods \(U_y\) 
of \(y\)  and \(U_z\)  of \(z\)  such that \(U_y \cup U_z\)  is homeomorphic to \(U_z\) . The neighborhoods
\(U_y\)  give a covering of \(V_2-W_2\) , hence there is a finite sub-covering. By making the neighborhoods
smaller, we can assume that they are disjoint. Since each neighborhood \(U_y\)  can be absorbed into \(U_z\) ,
the set \(V_2-W_2\)  can be absorbed into \(E-(V_1 \cup V_2)\)  and hence, \(E-(V_1 \cup V_2)\)  is
homeomorphic to \(E-(V_1 \cup W_2)\) .
Now consider surfaces \(\Sigma _{V_i}\)  and \(\Sigma _{W_i}\) 
whose end points are \(V_i\)  and \(W_i\) . We can choose these surfaces so that \(\Sigma _{W_i} \subset \Sigma _{V_i}\) 
and so that \(\Sigma _{V_i}\)  is disjoint from \(\Sigma _{V_{3-i}}\) . Also, we can assume
\(\Sigma _{V_i} - \Sigma _{W_i}\)  and \(\Sigma -(\Sigma _{W_1} \cup \Sigma _{W_2})\)  all have
genus zero or infinity. The end space of \(\Sigma _{V_i}- \Sigma _{W_i}\) 
is homeomorphic to the end space of \(\Sigma -(\Sigma _{W_1} \cup \Sigma _{W_2})\) 
and they have the same
genus. Also \(\Sigma _{V_1}\)  and \(\Sigma _{W_1}\)  have homeomorphic end spaces and the same
genus. Hence, there is a map \(h_1\)  such that \(h_1(\Sigma _{W_1})=\Sigma _{V_1}\)  and
\(h_1(\Sigma _{V_2}) = h_1(\Sigma _{V_2})\) . The map \(h_2\)  in Definition REF  can
also be similarly constructed.
The construction of the maps \(f_i\)  follows similarly. As above, we can send
\(\Sigma _{W_1}\)  to \(\Sigma -\Sigma _{V_2}\)  while fixing some neighborhood
\(W_2^{\prime }\)  of \(x_2\)  as long as \(W_2^{\prime }\)  is small enough so that \(V_2-W_2^{\prime }\)  intersect \(E(z)\)  for every
\(z \in \mathcal {M}(E-\lbrace x_1, x_2\rbrace )\) . This proves that \(\Sigma \) 
is telescoping and hence \(\operatorname{Map} (\Sigma )\)  is CB. Note that since
\(\operatorname{FMap} (\Sigma )\)  and \(\operatorname{Map} (\Sigma )\)  are quasi-isometric this means
\(\operatorname{FMap} (\Sigma )\)  is also CB which proves the theorem.
